I'm sure other people will have better answers than mine, but I'll give it a shot:

The 19th century, especially in Europe, was a time of incredible ideological growth and dramatic political upheaval. The political philosophies we hear so much about today, socialism, communism, anarchism (and even liberalism, to an extent) were born and developed during this time. Despite revolutionary unrest (most notably in 1848) large monarchies remained in control of Europe. This map is useful for visualizing how powerful monarchies still were. At the beginning of 1871, France was the only large republic in Europe, and even they had not completely given up looking for a king.

If you were a left-wing thinker at this time, it probably looked pretty grim. The revolutions of 1848 were exciting, but they demonstrated the difficulties of overthrowing a great power. There was a lot of disagreement as to how leftists (broadly speaking) should move forward. For simplicity's sake, you can sort them into two camps: electoralists, who thought the Left could effect change by moving through current institutions of the state (especially in relatively new parliamentary monarchies, such as the German Empire) and insurrectionists, who believed that any real change would be impossible without the overthrow of current regimes. In 1871, thought on the subject was based mostly on the events of the French revolution and the 1848 revolutions. This is what made the Paris Commune so exciting to leftists. Though relatively short-lived, the events of the Commune provided valuable, real-world demonstrations of the phenomena people were writing about at the time. Leftists drew various conclusions from the Commune. Some believed that it showed the futility of insurrectionism, while some thought the exact opposite. In the same way that people use presidential elections in the US today as fodder to credit or discredit ideas or political philosophies, the Paris Commune was viewed as an undeniably important example of a real revolutionary moment, and writers like Marx, Bakunin, and Kropotkin seized the opportunity to both learn from the events and to justify their specific brands of leftism.

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Sources:

Daggers, Rifles and Dynamite: Anarchist Terrorism in Nineteenth Century Europe,” Terrorism and Political Violence Volume 16, Number 1 (2004): 116-153

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A